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IB Maths AI HL · Unit 4: Statistics and Probability

IB Maths AI HL Random Variables and Poisson Questions

Updated October 2026

Exam-style IB Maths AI HL random variables and Poisson questions with worked solutions. Start with the 3 fully worked examples below — each is solved step by step, showing the working that earns the marks — then try the practice questions and check your working against the mark scheme.

47 questions
Paper 1: 19
Paper 2: 28
13 easy
19 medium
15 hard
3 worked examples

Practise Random Variables and Poisson questions →

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Start with one question on discrete random variables. Stuck? Take a hint. No sign-in needed.

Find

E (X)

for the distribution in the table.

$x$	1	2	3	4
$P (X = x)$	$\frac{1}{10}$	$\frac{1}{5}$	$\frac{3}{10}$	$\frac{2}{5}$

Choose your answer · keys 1–4

1

10

3

2.5

$x$	0	1	2
$P (X = x)$	$\frac{1}{4}$	$\frac{1}{2}$	$\frac{1}{4}$

Find

E (X)

for the distribution in the table.

$x$	2	5	10
$P (X = x)$	$\frac{1}{2}$	$\frac{3}{10}$	$\frac{1}{5}$

Choose your answer · keys 1–4

\frac{17}{3}

29.5

4.5

1

What you need to know

Poisson for rare events (call-centre arrivals, defects per unit), normal for continuous measurements. HL AI Paper 3 combines both. Normal and Poisson distributions overview →

What's examined in AI HL random variables and Poisson

The question bank covers these random variables and Poisson question types (number of questions in brackets):

Poisson Distribution Modelling (24)
Probability and Distribution Properties (16)
Expected Value and Variance Rules (7)

Key formulas

Mean & variance of Poisson: $E (X) = λ, Var (X) = λ$
Poisson distribution: $P (X = r) = \frac{e^{- λ} λ^{r}}{r!}$

In the same notation as the IB formula booklet. All AI HL formulas →

Random Variables and Poisson worked examples

Worked example 1: Calculating Basic Poisson Probabilities · easy

The number of emails received by a small business support desk follows a Poisson distribution with an average rate of $4.5$ emails per hour. Find the probability that the desk receives exactly $6$ emails in a randomly chosen hour.

Solution

6 steps · 1 to fill in

Examiner tip: Use `Poisson Ppd` (Point) for exact values like

P (X = 6)

, and use `Poisson Pcd` (Cumulative) for range values like

P (X \leq 6)

Worked example 2: Scaling the Poisson Mean · medium

A factory machine breaks down at an average rate of $0.8$ times per week. Assuming the breakdowns follow a Poisson distribution, calculate the probability that the machine breaks down exactly $4$ times in a $4$ -week period.

Solution

6 steps · 1 to fill in

Examiner tip: The defining characteristic of the Poisson distribution is that the mean (

λ

) is strictly proportional to the size of the interval (time, area, or volume) being observed.

Worked example 3: Summing Independent Poisson Distributions · hard

On a quiet rural road, the number of cars passing per hour follows a Poisson distribution with mean $2.5$ . The number of trucks passing per hour follows an independent Poisson distribution with mean $1.2$ . Find the probability that a total of exactly $5$ vehicles (cars and trucks combined) pass by in a randomly chosen one-hour period.

Solution

7 steps · 1 to fill in

Examiner tip: You can only add the means (

λ

) to form a new Poisson distribution if the individual events are completely independent of each other.

Try these IB Maths AI HL random variables and Poisson questions

Three questions from the bank, easiest first. Mark schemes and AI marking of your written working are in the practice area. One of them has its full worked solution here, open to everyone.

Question 1 · easy · 4 marks · Paper 1

A random variable $X$ has an expected value $E (X) = 8$ and a variance $V a r (X) = 5$ .
A new variable is defined by the linear transformation $Y = 3 X - 4$ .
Calculate $E (Y)$ and $V a r (Y)$ .

Attempt it and see the mark scheme →

Question 2 · medium · 5 marks · Paper 1

The weights of apples are normally distributed with a mean of $150$ g and a standard deviation of $10$ g.
If two apples are chosen at random, find the probability that their combined weight is greater than $315$ g.

Full worked solution Free, no sign-in

Let $A$ be an apple. $A_{1} \sim N (150, 10^{2})$ and $A_{2} \sim N (150, 10^{2})$ .
Let $T = A_{1} + A_{2}$ .
$E (T) = 150 + 150 = 300$ . [M1]
$V a r (T) = 10^{2} + 10^{2} = 100 + 100 = 200$ . [M1][A1]
Standard Deviation of $T = \sqrt{200} \approx 14.14$ . [A1]
Find $P (T > 315)$ using Normal CDF on GDC with mean $300$ and SD $\sqrt{200}$ .
$P (T > 315) \approx 0.144$ (3 s.f.). [A1]

Every question here has a full worked solution. See them all →

Question 3 · hard · 7 marks · Paper 1

Let $X$ be the volume of juice in a bottle, where $E (X) = 500$ ml and $V a r (X) = 15$ .
A student claims that taking one bottle and multiplying its volume by two, $Y = 2 X$ , has the exact same variance as taking two independent bottles and adding their volumes together, $W = X_{1} + X_{2}$ .
Determine algebraically whether the student is correct, and explain the mathematical reason for the difference in variance.

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FAQ

How many IB Maths AI HL random variables and Poisson questions are there?

There are 47 exam-style random variables and Poisson questions in the AI HL question bank (Paper 1: 19 · Paper 2: 28), graded 13 easy, 19 medium, 15 hard. Every question has a full IB-style mark scheme (M, A and R marks).

Is random variables and Poisson on Paper 1 or Paper 2?

Both. In the bank, Paper 1: 19 · Paper 2: 28. Practise with your GDC — AI papers expect calculator methods throughout.

Where can I get the mark schemes?

Open the AI HL Unit 4 practice page: every question has a step-by-step IB-style mark scheme, and you can photograph your working for instant AI marking. The worked examples on this page are free.

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Teaching this topic?

Free ready-to-teach lessons (slides and teacher notes): Poisson distribution · Exponential distribution (extension).

IB Maths AI HL Random Variables and Poisson Questions

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What you need to know

What's examined in AI HL random variables and Poisson

Key formulas

Random Variables and Poisson worked examples

Worked example 1: Calculating Basic Poisson Probabilities · easy

Solution

Worked example 2: Scaling the Poisson Mean · medium

Solution

Worked example 3: Summing Independent Poisson Distributions · hard

Solution

Try these IB Maths AI HL random variables and Poisson questions

Question 1 · easy · 4 marks · Paper 1

Question 2 · medium · 5 marks · Paper 1

Full worked solution Free, no sign-in

Question 3 · hard · 7 marks · Paper 1

Revise random variables and Poisson for the exam

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